 Open Access
 Total Downloads : 1819
 Authors : Prof.Sable K.S., Miss Patil Archana A.
 Paper ID : IJERTV1IS6401
 Volume & Issue : Volume 01, Issue 06 (August 2012)
 Published (First Online): 30082012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Optimization Of Retaining Wall By Using Optimtool In MATLAB

Prof.Sable K.S., PHD, Principal in Sayadri College of Engg.Alephata, Pune

Miss Patil Archana A. M.E.Structure in Amrutvahini College of Engg.,Sangamner,Pune
Abstract
Optimization of concrete retaining walls is an important task in geotechnical and structural engineering. However other than stability considerations very often in such design the settlement aspects is neglected. As such, attention to various aspects of geotechnical engineering design needs to be considered. However, consideration of all these aspects makes the design complicated. To economize the cost under such situation needs to vary the dimensions of the wall several times making it very tedious and monotonous. As it is extremely difficult to obtain a design satisfying all the safety requirements, it is necessary to cast the problem as one of the mathematical non linear programming techniques. A program is developed for analysis and designing low cost or lowweight cantilever reinforced concrete retaining walls with and without base shear key in matlab for optimtool. The optimtool is used to find the minimum cost and weight for concrete retaining walls. Illustrative cases of retaining wall are solved, and their results are presented and discussed by using Interior point method from optimtool. The optimum design formulation allows for a detailed sensitivity analysis to be made for variation in top thickness of stem, surcharge load and internal angle of friction with different height.

Introduction
Optimization is the act of obtaining the best result under given circumstances. In design process, engineers have to take many technological and managerial decisions at several stages. The ultimate goal of all such decisions is either to minimize the effort required or to maximize the desired benefit. The objective of the optimization is to minimize the total cost or total weight per unit length of the retaining structure subjected to constraints based on stability, bending moment and shear force capacities, and the requirements of the IS 4562000. The improvements in
numerical methods and computer technology have given impetus to this concept of optimization.

Methodology
Several studies have been done to develop methodologies for the analysis and design of cantilever retaining walls. However, limited work has been undertaken to develop methods for their optimum cost design. In general, the forces acting on this model of a retaining structure are consistent; formulation includes both passive forces on the front of the toe and base shear key sections and the bearing force of the base soil. Figure1 shows the general forces acting on the retaining wall: WW is the combined weight of all the sections of the reinforced concrete wall; WS is the weight of backfill acting on the heel; WT is the weight of soil on the toe; Q is the surcharge load; PA is force due to the active earth pressure; PK and PT are the forces due to passive earth pressure on the base shear key and front part of the toe section, respectively; and PB is the force due to the bearing stress of the base soil. Three failure modes are considered in the analysis of the retaining structure: overturning, sliding, and bearing stress. The overturning moment about the toe of the wall is a balance of the force due to the active soil pressure of the retained soil weight and the self weight of the concrete structure, the soil above the base, and the surcharge load. The passive forces on the front of the toe and the base shear key section are not considered in the overturning moment. The factor of safety for overturning FSO about the toe is defined as:
MR
FSO =
MO
where MR is the sum of the moments about toe resisting overturning and MO is the sum of the moments about toe tending to overturn the structure.
the sum of the vertical forces (due to the weight of wall, the soil above the base, and surcharge load), and E is the eccentricity of the resultant force system expressed as:
B M R MO
E = –
2 V
The eccentricity is determined from the ratio of the summation of overturning moments about the toe to the sum of vertical forces. The factor of safety for the bearing capacity FSB is:
qu
FSB =
qmax
Figure 1. "General Forces Acting on the Retaining Wall"
The active earth pressure coefficient ka is: 1 sinÃ˜
KP =
1+ sinÃ˜
where Ã˜ is the angle of internal friction. The passive earth pressure coefficient KP is:
1+sinÃ˜
KP =
1 sinÃ˜
For the sliding mode of failure, only the horizontal component of the active force is considered. Horizontal resisting forces are due to the weight of wall and soil on the base, surcharge load, friction between soil and base of wall, and passive force due to soil on the toe and base shear key sections. The factor of safety against sliding FSS is defined as:
FR FSS =
FD
where FR is the sum of the horizontal resisting forces and FD is the sum of the horizontal sliding forces.
In the bearing analysis of the structure, the base of retaining wall is considered to be a shallow foundation. The minimum and maximum applied bearing stresses on the base of the foundation are:
V 6*E
qMax = 1 Â±
Min B B
where qmin and qmax are the bearing stresses on the toe and heel sections, B is the width of the base, V is
where qu is the ultimate bearing capacity of the foundation.

Formulation of Optimum Design Problem
The general three phases considered in the optimum design of any structure are: structural modelling, optimum design modelling, and the optimization algorithm. In the structural modelling, the problem is formulated as the determination of a set of design variables for which the objective of the design is achieved without violating the design constraints. For the optimum design modelling, study the problem parameters in depth, so as to decide on design parameters, design variables, constraints, and the objective function. In the search for finding optimum design starts from a design or from a set of designs to proceed towards optimum. For economic design of retaining wall, optimization methodology and above parameters are discussed in the following sections.
Structural Modelling
In optimal design problem of retaining wall the aim is to minimize the construction cost and weight of the wall under constraints. This optimization problem can be expressed as follows:
minimize f(X) subject to
gi (X) 0 i=1,2, . . . . p
hj (X) = 0 j=1,2,. . . . m
Lk Xk Uk k=1, 2 . . . n
where f(X) is the objective function gi(X), hj(X) are inequality and equality constraints respectively and Lk, Uk are lower and upper bound constraints. To economic design of retaining wall, the objective function, design variables and design constraints should be defined explicitly.
Optimum Design Modelling

Design Variables
The design variables are divided into two categories: those that prescribe the geometric dimensions of wall crosssection, and those that model the steel reinforcement. In general, there are eight geometric design variables representing the dimensions of the retaining wall: X1 is the width of the base, X2 is the toe projection, X3 is the thickness at the bottom of the stem, X4 is the thickness at the top of the stem, X5 is the thickness of base slab, X9 is the distance from toe to the front of the base shear key, X10 is the width of the key, and X11 isthe depth of the key. There are four additional design variables related to the steel reinforcement of the various sections of the retaining wall: X6 is the vertical steel reinforcement in the stem, X7 is the horizontal steel reinforcement in the toe, X8 is the horizontal steel reinforcement in the heel, and X12 is the vertical steel reinforcement in the base shear key. While the geometric design variables may be either continuous or discrete values, the steel reinforcement design variables are modeled as a set of discrete values. In this formulation, where the retaining structure is designed for a unit length, the number of bars in a unit meter length of the retaining wall conforms to the minimum and maximum amount of steel allowed.
Figure 2. "Mathematical Model used for Optimum Design of Reinforced Concrete Cantilever Retaining Wall"

Constraints
The typical design philosophy for retaining structures seeks designs that provide safety and stability against
failure modes and comply with concrete building code requirements. These requirements may be classified into four general groups of design constraints: stability, capacity, reinforcement configuration, and geometric limitations. Each of the design constraints are posed as penalties on the overall objective functions of the design and are nonzero only when violated. In other words, if the design is feasible, the sum of the constraint penalties will be zero. According to IS code 456:2000, the design constraints may be classified as geotechnical and structural requirements. These requirements represent the failure modes as a function of the design variables. Feasible retaining wall designs should provide minimum factor of safety coefficients for overturning, sliding, and bearing capacity failure modes. Failure modes is summarized in table are as follows
Table1. "Failure Modes of Retaining Wall"
Inequality constraints
Failure mode
g1(X)
Overturning stability
g2(X)
Maximum bearing capacity
g3(X)
Minimum bearing capacity
g4(X)
Sliding stability
g5(X)
No tension condition
g6(X)
Moment at bottom of stem
g7(X)
Moment at toe
g8(X)
Moment at heel
g9(X)
Moment at shear key
g10(X)
Shear at bottom of stem
g11(X)
Shear at Toe
g12(X)
Shear at heel
g13(X)
Shear at shear key
g14(X)
Minimum area reinforcement criteria
g15(X)
Maximum area reinforcement criteria
g16(X)
Additional geometric constraints

Overturning failure mode The stabilizing moments, due to vertical forces must be greater than the overturning moments, due to horizontal forces to prevent rotation of the wall around its toe. The stabilizing moments result mainly from the selfweight of the structure, whereas the main source of overturning moments is the active earth pressure. Overturning failure is a result of excessive lateral earth pressures with relation to retaining wall resistance thereby
causing the retaining wall system to topple or rotate (overturn).
g1(X) = FSo – (Mvtotal/Mhtotal) 0
where Mvtotal = Total vertical moment of forces that tends to resist overturning about toe.
Mhtotal = Total horizontal moment of forces that tends to overturn about toe.
FSo = Factor of safety against overturning.

Bearing failure mode The bearing capacity of the foundation must be large enough to resist the stresses acting along the base of the structure.
g2(X) = Pmax – S.B.C 0
where S.B.C. = Safe bearing capacity of soil Pmax=Maximum contact pressure at the interface between the wall structure and the foundation soil. g3(X) = Pmin 0
Pmin = Minimum contact pressure at the interface between the wall structure and the foundation soil.

Sliding failure modeThe net horizontal forces must be such that the wall is prevented from sliding along its foundation. The most significant sliding force component usually comes from the lateral earth pressure acting on the active (backfill) side of the wall. Sliding failure is a result of excessive lateral earth pressures with relation to retaining wall resistance thereby causing the retaining wall system to move away (slide) from the soil it retains.
g4(X) = FSs – ((Vtotal*Âµ+Horizontal force from passive pressure)/Htotal) 0
where (Vtotal*Âµ+Horizontal force from passive pressure)=Resistance to sliding
Htotal=Total horizontal driving forces. FSs=Factor of safety against sliding.

Tension failure mode For stability, the line of action of the resultant force must lie within the middle third of the foundation base.
g5(X) = E – (B/6) 0
Where B = Base width of the wall
E = Eccentricity of the resultant force.

Moment failure mode The maximum bending moment at the face of the Support should be less than the resistance moment of stem:
The flexural strength Mrs is calculated as: Mrs = 0.87 *As* fy* (ds 0.416 *Xu)
Where Xu is the location of neutral axis for provided steel,
Xu = (0.87*fy*As)/ (0.36*fck*b)
As is the crosssectional area of steel reinforcement at stem, fy is the yield strength of steel,
ds is the effective depth at stem. g6(X) = Ms Mrs 0
Where Mrs = Flexural strength of stem
Ms = Maximum bending moment at the face of the wall A critical section for the moment is considered at the junction of stem with toe slab. So maximum bending moment at a vertical section at the junction of the stem with toe slab should be less than the moment of resistance of toe slab:
g7(X) = Mt Mrt 0
Where Mrt = Flexural strength of the toe slab
Mt = Maximum bending moment at a vertical at the junction of the stem with toe slab.
A critical section for the moment is considered at the junction of stem with heel slab. So maximum bending moment at a vertical section at the junction of the stem with heel slab should be less than the resistance moment of the heel slab:
g8(X) = Mh Mrh 0
Where Mrh = Flexural strength of the heel slab
Mh= Maximum bending moment at a vertical at the junction of the stem with heel slab.
g9(X) = Mk Mrk 0
where Mk = Moment at base shear key
Mrk = Maximum bending moment at base shear key.

Shear failure mode – The retaining wall has to be designed as a cantilever slab to resist moments and shear forces.
g10(X) = Vs Vus 0
Where Vus = Shear capacity of concrete at stem As = Area of reinforcement at stem.
ds = Effective depth at stem.
Vs = design shear carrying capacity at stem
The net loading acts upwards and flexural reinforcement has to be provided at the bottom of the toe slab. To prevent toe shear failure, nominal shear stress at the junction of stem with toe slab should be less than shear strength of concrete at toe.
g11(X) = Vt Vut 0
Where Vut = Shear capacity of concrete at toe At = Area of reinforcement at toe.
dt = Effective depth at toe.
Vt = Design shear carrying capacity of toe.
The net loading acts downwards and flexural reinforcement has to be provided at the top of the heel slab. To prevent toe shear failure, critical section for the shear is considered at the junction of stem with heel slab and nominal shear stress at the junction of stem with heel slab should be less than shear strength of concrete.
g12(X) = Vh – Vuh 0
Where Vuh = Shear capacty of concrete at toe Ah = Area of reinforcement at heel.
dh = Effective depth at heel.
Vh = design shear carrying capacity of toe.
Variables"
Note h= Height of stem in m
Design variables
Lower bounds
Upper bounds
Width of the base X1
X1=0.4*h*(12/11)
X1=(0.7*h)/0.9
Toe projection X2
X2=[0.4*h*(12/11)]/3
X2=[(0.7*h)/0.9]/3
Thickness at the
bottom of the stem X3
X3=0.2
X3=(h/0.9)/10
Thickness at the top
of the stem X4
X4=0.2
X4=0.2
Thickness of base
slab X5
X5=[h*(12/11)]/12
X5=(h/0.9)/10
The vertical steel reinforcement in the stem, per unit length
of wall X6
X6=0.0012* X3
X6=0.04*X3
Horizontal steel reinforcement in the
toe, per unit length of wall X7
X7=0.0012*X5
X7=0.04*X5
Horizontal steel reinforcement in the heel, per unit length
of wall X8
X8=0.0012*X5
X8=0.04*X5
The distance from toe to the front of the
base shear keyX9
X9=0.4*h*(12/11)
X9=(0.7*h)/0.9
Width of the shear key X10
X10=0.3
X10=(h/0.9)/10
Depth of the shear
keyX11
X11=0.3
X11=(h/0.9)/10
The vertical steel reinforcement in the
base shear keyX12
X12=0.0012*X10
X12=0.04*X3
g (X) = Vk – Vuk 0
13
Where Vuk=Shear capacity of concrete at shear key Ak = Area of reinforcement at shear key.
dk = Effective depth at shear key.
Vk = design shear carrying capacity of shear key.

Minimum area of reinforcement criteria g14(X) = (0.12/100)*b*DsAs 0
b = Base width of retaining wall
Ds = Thickness at the bottom of stem As = Area of reinforcement in stem g15(X) = (0.12/100)*b*D At 0
D = Thickness of the base slab At = Area of reinforcement in toe
g16(X) = (0.12/100)*b*D Ah 0
Ah = Area of reinforcement in heel

Maximum area of reinforcement criteria g17(X) = As (4/100)*b*Ds 0
g18(X) = At (4/100)*b*D 0 g19(X) = Ah (4/100)*b*D 0

Additional geometric criteriaThere is several additional geometric constraints that are applied to combinations design variables to prevent infeasible retaining wall dimensions.
g20(X) = X2+X3 X1 0 g21(X) = X9+X10X1 0

Lower and upper bound constraints
The derived constraint expressions are found to be highly nonlinear in the design variables.


Objective Function

The objective function is a function of design variables the value of which provides the basis for choice between alternate acceptable designs. The objective of
Table2." Lower and upper bounds of design
design may be minimization of weight /cost/stress concentration factor. In structural designs the objective function is usually weight or cost minimization. The forms of the two objective functions for this optimization are consistent, i.e. the cost of concrete and reinforcing steel (both include the cost of the material per unit volume and costs associated with labour and installation). The cost function f (cost) is:
f (cost) = Cs*Wst + Cc*Vc
where Cs is the unit cost of steel, Cc is the unit cost of concrete, Wst is the weight of steel per unit length of the wall, and Vc is the volume of concrete per unit. length of the wall. The second objective function is based solely on the weight of the materials. The weight function f(weight) is:
f(weight) = Wst + 100 *Vc *c
where c is the unit weight of concrete and a factor of 100 is used for consistency of units.
Table3."Optimum Values of Design Variables for 3.2m Height Retaining Wall"
Design variabl es 
Unit 
Lower Bounds 
Upper Bounds 
Optimum values Minimum Cost 
Optimum values Minimum Weight 
X1 
m 
1.396 
2.488 
1.7594 
1.7569 
X2 
m 
0.466 
0.8296 
0.5979 
0.6291 
X3 
m 
0.2 
0.356 
0.301 
0.201 
X4 
m 
0.2 
0.2 
0.2 
0.2 
X5 
m 
0.291 
0.356 
0.291 
0.2730 
X6 
m2 
0.00024 
0.0142 
0.000519 
0.000884 
X7 
m2 
0.000349 
0.0142 
0.000349 
0.000328 
X8 
m2 
0.000349 
0.0142 
0.000349 
0.000335* 
Table4."Values of behavioural Constraints at Optimum values of Design Variables for 3.2 m retaining Wall"
Symbol 
Minimum Cost 
Minimum Weight 
g1(X) 
0.7214 
0.7281 
g2(X) 
146.2108 
146.4938 
g3(X) 
0.0020 
0.0002 
g4(X) 
0.2610 
0.2486 
g5(X) 
0.0000 
0.0000 
g6(X) 
0.2649 
0.3602 
g7(X) 
19.8976 
13.3762 
g8(X) 
8.1518 
0.2704 
g10(X) 
49.1923 
52.8123 
g11(X) 
10.6466 
2.6787 
g12(X) 
18.5880 
12.9409 
Objective Function 
Unit 
Optimum value 
Weight of Steel (kg/m) 
Volume of Concrete (m3/m) 
Min. cost 
Rs/m 
13585 
51.4396 
1.3136 
Min.weight 
kg/m 
2883.2 
80.1159 
1.1212 
Table5."Input Parameter for design example"
Fck
Input Parameter 
Unit 
Symbol 
Design Value 

For 3.2m 
For 6.3m 

Internal angle of friction 
Degree 
Phi 
35 
35 
Surcharge load 
kN/m2 
Q 
10 
10 
Backfill slope 
Degree 
beeta 
0 
0 
Height of stem 
m 
H 
3.2 
6.3 
Yield strength of steel 
kN/m2 
Fy 
500*103 
500*103 
Characteristic strength of concrete 
kN/m2 
25*103 
25*103 

Density of soil 
kN/m3 
rhosoil 
17 
17 
Unit weight of concrete 
kN/m3 
D 
25 
25 
Concrete cover 
m 
cover 
0.025 
0.025 
Safe bearing capacity of soil 
kN/m2 
S.B.C. 
250 
250 
Coefficient of friction under base 
mue 
0.55 
0.55 

Factor of safety for overturning stability 
FSo 
1.4 
1.4 

Factor of safety against sliding 
FSs 
1.4 
1.4 

Factor of safety for bearing capacity 
FSb 
3 
3 

Cost of steel 
Rs/kg 
Cs 
60 
60 
Cost of concrete 
Rs/m3 
Cc 
8000 
8000 
% minimum steel 
% 
min 
0.12 
0.12 
% maximum steel 
% 
max 
4 
4 
Table6."Optimum Values of Objective Functions, Weight of steel, and Volume of Concrete for 3.2 m retaining Wall"
Example1 Optimization for 3.2m Height Cantilever Retaining Wall
In general relation between linear and nonlinear is found to be difficult. The numbers of cases of retaining wall are optimized for one meter length of retaining wall. The design variables described as X1 is base width of retaining wall, X2 is toe width of retaining wall, X3 is bottom thickness of stem, X4 is top thickness of stem, X5 is base thickness of retaining wall, X6 is vertical reinforcement area provided in stem, X7 is horizontal steel area in toe of retaining wall, and X8 is horizontal reinforcement area in heel of the retaining wall. As shown in Table 3, out of first five design variables that describe the shape of the optimum retaining wall and remaining three describe the area of reinforcement in stem, toe, and heel respectively. For a specified set of design parameters used more iterative method from optimtool i.e. interior point method the derived constraints expressions are found to be highly nonlinear in the design variables. In addition, all the design variables have lower bounds and upper bounds, which are considered as per given by Saribus and Erabatur [1] and IS 4562000.In analysis process design considerations are related to total height of retaining wall but in optimization, taking initial assumptions i.e. lower bound and upper bound of the design variables are related to the height of stem and thickness of base is as design variable.. Lower bound and upper bound constraints are considered for all examples as per Table 2
Example2 – Optimization for 6.3m Retaining Wall without Shear Key
In optimization of 6.3m retaining wall, changing height of retaining wall in program coding and in objective function, as per diameter of steel bar obviously changed weight of steel. From constraint equations and Table 7 gives idea, the vertical reinforcement area of stem(X6) is depends on Bottom width of stem(X3).In cost optimization model shows that X3 increases with X6 and vice versa in weight optimization model. In weight optimization model volume of concrete reduces and weight of steel increases than that of cost optimization model.
Table7."Optimum Values of Design Variables for 6.3m Height Retaining Wall without Shear Key"
Design variable 
Unit 
Lower Bounds 
Upper Bounds 
Optimum values Minimum Cost 
Optimum values Minimum Weight 
X1 
m 
2.749 
4.90 
3.3008 
3.3108 
X2 
m 
0.916 
1.633 
1.0305 
1.1057 
X3 
m 
0.2 
0.70 
0.6284 
0.3060 
X4 
m 
0.2 
0.2 
0.2 
0.2 
X5 
m 
0.573 
0.70 
0.5910 
0.5910 
X6 
m2 
0.00024 
0.028 
0.0015 
0.004639 
X7 
m2 
0.000688 
0.028 
0.0007092 
0.0007092 
X8 
m2 
0.000688 
0.028 
0.001139 
0.001252 
Table8."Values of behavioural Constraints at Optimum values of Design Variables for 6.3m Height Retaining Wall without Shear Key"
Symbol 
Minimum Cost 
Minimum Weight 
g1(X) 
0.7031 
0.7252 
g2(X) 
61.246 
58.6384 
g3(X) 
0.0050 
0.0050 
g4(X) 
0.1370 
0.1630 
g5(X) 
0.0000 
0.000 
g6(X) 
2.0534 
0.0175 
g7(X) 
106.07 
97.2584 
g8(X) 
68.3736 
34.0904 
g10(X) 
85.302 
133.7170 
g11(X) 
28.147 
17.9790 
g12(X) 
0.0243 
0.0028 
Table9." Optimum values of Objective Functions, Weight of Steel, and Volume of Concrete"
Objective Function 
Unit 
Optimum value 
Optimum Weight of Steel(kg/m) 
Optimum Volume of Concrete (m3/m) 
Min. cost 
Rs/m 
52589 
269.296 
4.5602 
Min. weight 
kg/m 
9615.8 
739.390 
3.5506 
Example3 Optimization for 6.3m Height Retaining Wall with Shear Key
In this set of retaining wall designs, a base shear key is included in the design variables. Four
additional variables are added to optimization program, X9 is the location of shear key from toe, X10 is width of shear key, X11 is height of the shear key and X12 is vertical reinforcement area provided in shear key. In this program, additional geometric constraints and moment capacity, shear capacity constraints are included. In objective function equations included weight of steel for key with respective length and spacing between the reinforcement and volume of concrete required for shear key with respective to design variables. Provision of shear key for greater height is best solution.
Objective Function 
Unit 
Optimum Value 
Optimum Weight of steel(kg/m) 
Optimum Volume of concrete (m3/m) 
Min. cost 
Rs/m 
52262 
264.153 
4.5516 
Min.weight 
kg/m 
9652.73 
717.322 
3.742 
Table10." Optimum Values of Objective Functions, Weight of Steel, and Volume of Concrete for 6.3m Height Retaining Wall with Shear Key"

Sensitivity Analysis
A sensitivity analysis adds quality to a design and supplies very important information on the work being designed from the view point of cost and reliability. The sensitivity analysis is very useful to (a) the designer, who can know which data values are more influential on the design, (b) to the builder, who can know how changes in prices influence the total cost, and (c) to the code maker, who can know the costs and reliability changes associated with an increase or decrease in the required safety factors or failure probabilities. The design parameters considered in above cases having wide range of parameters that are related to loading, geometry, soil properties, code specifications, unit cost, and other characteristics of construction materials. Sensitivity of the optimum solution to changes in these parameters is an important issue as far as practical design is concerned. The analysis results include the sensitivities of the optimum weight and optimum cost as objective functions and the Table11." Optimization for 6.3m Height Retaining
Wall with Shear Key"
Table12." Values of behavioural Constraints at Optimum values of Design Variables for 6.3m Height Retaining Wall with Shear Key"
Design variables
unit
Lower bounds
Upper bounds
Optimum values minimum
cost
Optimum values minimum
weight
X1
m
2.749
4.90
3.295
3.3044
X2
m
0.916
1.633
1.0418
1.1046
X3
m
0.2
0.7
0.6170
0.3050
X4
m
0.2
0.2
0.2
0.2
X5
m
0.573
0.70
0.5730
0.5730
X6
m2
0.00024
0.028
0.001524
0.004692
X7
m2
0.000688
0.028
0.000688
0.000688
X8
m2
0.000688
0.028
0.001169
0.001283
X9
m
2.749
4.9
1.5
1.5
X10
m
0.3
0.7
0.3
0.3
X11
m
0.3
0.7
0.3
0.3
X12
m2
0.00024
0.028
0.000360
0.000360
Symbol
Minimum Cost
Minimum Weight
g1(X)
0.6976
0.7225
g2(X)
62.8021
59.5761
g3(X)
0.0263
0.0003
g4(X)
0.3878
0.4233
g5(X)
0.0002
0.0000
g6(X)
0.1085
0.0144
g7(X)
95.4298
87.3916
g8(X)
62.5473
32.7742
g9(X)
40.7251
40.7251
g10(X)
84.8838
134.8692
g11(X)
23.4356
13.5859
g12(X)
0.0083
0.0568
g13(X)
76.724
76.7238
optimum values of the several design variables. As a representative of such analyses, results concerned with the sensitivity of optimum solutions with respect to height and top thickness of stem, surcharge load, internal angle of friction of retained soil. In sensitivity analysis, sensitivity of the objective functions explained for all design parameters which are considered in number of above not cases. Both objective functions are considered and their optimum
value variations with respect to changes in height of the stem for various top thickness values are given in Grappand 2. In the range of the stem height (36.5m) shows that for higher values of height, the optimum cost and weight become more sensitive to variations in the stem height. This is more apparent for the cost function. For example, when top thickness of stem (X4)=0.2m and height changes from 3.0to 6.5m, the cost of the wall increases 3.6524 times, whereas the weight increases 3.0 times. Smaller top thickness values of the stem produce more favourable optimum solutions for both objective functions. From table, shows that the optimum values for a given height, the first four design variables are not much affected by increases of X4, from X4=0.20 to0.30. For both minimization models, the optimum values of the last three design variables corresponding to reinforcing steel areas show sensitivity to changes in height, but in general to shifts in X4. Only horizontal steel reinforcing area in the cost and weight minimization model is influenced by changes in X4.
Grapp."Comparison of Optimum Cost and Height of Stem for different Top Thickness of Stem"
Grapp." Comparison of Optimum Weight and Height of Stem for different Top Thickness of Stem"
3500
3000
Optimum weight (Kg/m)
2500
2000
1500
1000
500
0
Grapp." Comparison of Optimum Weight, Optimum Cost and Surcharge Load for 3.0 m Height of Stem"
The input equation in terms of height of stem and surcharge load are prepared for the study of sensitivity for surcharge loads corresponding to input parameter. The optimum values for the objective functions are shown in Graph 3 for the surcharge load varying from 0 to 50 kPa. In this, as q changes from 0
18000
16000
14000
12000
Optimum Cost
10000
8000
6000
4000
2000
0
to 50 kPa, the optimum cost increases 0.613 times, and the optimum weight increases by 0.23 times. The cost the surcharge load increases, in both of the optimization models. The optimum value of horizontal reinforcement area in heel (X8) increases with an increase in the surcharge load, yet after 10kPa in the cost and weight minimization model. Minimization model is more sensitive to variations in surcharge load compared to the weight minimization model. As for the sensitivity of the design variables, significant sensitivity is observed in base width(X1) to stem thickness at bottom (X3) and vertical reinforcement in stem(X6). These variables increase as the surcharge load increases, in both of the optimization models. The optimum value of horizontal reinforcement area in heel (X8) increases with an increase in the surcharge load, yet after 10kPa in the cost and weight minimization model.
Graph4." Comparison of Optimum Weight, Optimum Cost and Internal Angle of Friction for
3.0 m Height of Stem"
From graph 4, sensitivity analysis related to the internal friction angle of retained soil, prepared different constrained equations in terms of H and Ã˜ for finding active pressure and passive pressure. The minimum weight model is more sensitive in the range of Ã˜ is 28 to 36 degree when compared to the minimum cost model. As far as the design variables are concerned, in the cost minimization as well as weight minimization model Base thickness (X5) and toe steel area (X7) are insensitive to change in the internal angle of friction. The retaining wall base width (X1), toe width (X2), stem bottom thickness (X3), reinforcement area in stem (X6), heel reinforcement area (X8), are decreases with an increase in the angle of internal friction.

Conclusion
The purpose of optimization is to choose the best one of the many acceptable designs available. For achieving economy in conventional analysis optimization programming made by considering geometric, moment and shear constraints, getting optimum value for number of cases of retaining wall with and without shear key. Sensitivity of the optimum solutions with respect to surcharge load, internal angle of friction gives the idea about effect on geometric parameters of retaining wall related to geotechnical and
structural requirements.
References

Sariba, A. and Erbatur, F. (1996). Optimization and Sensitivity of Retaining Structures. Journal of Geotechnical Engineering, 122(8), 649656.

Kaveh A. and Abadi A. S. M. (2010). Harmony Search Based Algorithm for the Optimum Cost Design of Reinforced Concrete Cantilever Retaining Walls. International Journal of Civil Engineering, 4(8), 336357.

Charles V. Camp and Alper Akin "Design of retaining walls using big bangbig crunch optimization Optimum Design of Cantilever Retaining Walls"

Sivakumar babu & b. m. basha Title "inverse reliability based design optimization of cantilever retaining walls" Indian institute of science of

bangalore, india

M. Asghar Bhatti Title "Retaining Wall Design Optimization with MS Excel Solver" Department of Civil & Environmental Engineering, University of Iowa, Iowa City, IA

M. Ghazavi, V. Salavati ,Title "Sensitivity analysis and design of reinforced concrete cantilever retaining walls using bacterial foraging optimization algorithm"
International Journal of Engineering Research & Technology (IJERT)
ISSN: 22780181
Vol. 1 Issue 6, August – 2012